A223702 -id:A223702 - cp's OEIS frontend

A223701 Irregular triangle of numbers k such that prime(n) is the largest prime factor of k^2 - 1.

3, 2, 5, 7, 17, 4, 9, 11, 19, 26, 31, 49, 161, 6, 8, 13, 15, 29, 41, 55, 71, 97, 99, 127, 244, 251, 449, 4801, 8749, 10, 21, 23, 34, 43, 65, 76, 89, 109, 111, 197, 199, 241, 351, 485, 769, 881, 1079, 6049, 19601, 12, 14, 25, 27, 51, 53, 64, 79, 129, 131, 155

Offset: 1

T. D. Noe, Apr 03 2013

Note that the first number of each row forms the sequence 3, 2, 4, 6, 10, 12,..., which is A039915. The first 25 rows, except the first, are in A181447-A181470.

			Irregular triangle:
  {3},
  {2, 5, 7, 17},
  {4, 9, 11, 19, 26, 31, 49, 161},
  {6, 8, 13, 15, 29, 41, 55, 71, 97, 99, 127, 244, 251, 449, 4801, 8749}

Andrew Howroyd, Table of n, a(n) for n = 1..16223 (first 25 rows for primes up to 97)
Florian Luca and Filip Najman, On the largest prime factor of x^2-1, arXiv:1005.1533 [math.NT], 2010.
Florian Luca and Filip Najman, On the largest prime factor of x^2-1, Mathematics of Computation 80 (2011), 429-435. (Paper has errata that was posted on the MOC website.)
Filip Najman, List of Publications Page (Adjacent to entry number 4 are links with the data files for the first 25 rows (=16223 terms) of this sequence)

Rows 2..25 with complete data are: A181447, A181448, A181449, A181450, A181451, A181452, A181453, A181454, A181455, A181456, A181457, A181458, A181459, A181460, A181461, A181462, A181463, A181464, A181465, A181466, A181467, A181468, A181469, A181470.
Row 26 is A181568.
Cf. A039915 (first terms), A175607 (last terms), A181471 (row lengths), A379344 (row sums).
Cf. A223702, A223703, A223704 (related tables).

t = Table[FactorInteger[n^2 - 1][[-1,1]], {n, 2, 10^5}]; Table[1 + Flatten[Position[t, Prime[n]]], {n, 6}]

A285283 Number of integers x such that the greatest prime factor of x^2 + 1 is at most A002313(n), the n-th prime not congruent to 3 mod 4.

Original entry on oeis.org

1, 4, 9, 15, 22, 32, 41, 57, 74, 94, 120, 156, 192, 232, 278, 325, 381, 448, 521, 607, 704, 811

Offset: 1

Tomohiro Yamada, Apr 16 2017

In other words, x^2 + 1 is A002313(n)-smooth.
Størmer shows that the number of such integers is finite for any n.
a(n) <= 3^n - 2^n follows from Størmer's argument.
a(n) <= (2^n-1)*(A002313(n)+1)/2 is implicit in Lehmer 1964.
Luca 2004 determines all integers x such that x^2 + 1 is 100-smooth, which is pushed to 200 by Najman 2010.

D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57--69.
Florian Luca, Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19--24.
Filip Najman, Smooth values of some quadratic polynomials, Glas. Mat. 45 (2010), 347--355. Tables are available in the author's Home Page (gives all 811 numbers x such that x^2+1 has no prime factor greater than 197).
A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arithmetica 13 (1967-1968), 177--236.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2), 48 pp.

Equivalents for x(x+1): A145604.

Cf. A002313, A014442, A185389, A223702, A285282, A379346 (first differences).

a(13)-a(22) added by Andrew Howroyd, Dec 22 2024

A379350 Triangle read by rows in which row n lists numbers k such that the greatest prime factor of k^2 + 2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

Original entry on oeis.org

0, 1, 2, 4, 5, 22, 3, 8, 14, 19, 140, 7, 10, 24, 41, 58, 265, 707, 6, 13, 25, 32, 44, 63, 146, 184, 602, 3407, 21362, 11, 30, 52, 71, 112, 194, 298, 481, 503, 2695, 3433, 4991, 16, 27, 59, 70, 102, 113, 317, 500, 586, 1048, 2951, 3424, 4972, 8240, 12658, 83834, 686210, 1306066

Offset: 1

Andrew Howroyd, Dec 22 2024

For any prime p, there are finitely many x such that x^2 + 2 has p as its greatest prime factor.

			Irregular triangle begins:
   p | {k}
-----+------------------
   2 | {0}
   3 | {1, 2, 4, 5, 22}
  11 | {3, 8, 14, 19, 140}
  17 | {7, 10, 24, 41, 58, 265, 707}
  19 | {6, 13, 25, 32, 44, 63, 146, 184, 602, 3407, 21362}
  41 | {11, 30, 52, 71, 112, 194, 298, 481, 503, 2695, 3433, 4991}
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..915 (first 21 rows for primes up to 193)
Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Filip Najman, List of Publications Page (Adjacent to entry number 7 are links with a data file for rows 2..21 (=914 terms) of this sequence).

Cf. A033203, A379351, A379352 (first terms), A185397 (last terms), A379349 (row lengths).

Cf. A223701, A223702, A242488.

A379346 Number of integers of the form k^2 + 1 whose greatest prime factor is A002313(n), the n-th prime not congruent to 3 mod 4.

Original entry on oeis.org

1, 3, 5, 6, 7, 10, 9, 16, 17, 20, 26, 36, 36, 40, 46, 47, 56, 67, 73, 86, 97, 107

Offset: 1

Andrew Howroyd, Dec 22 2024

See A223702 for additional information.

Florian Luca, Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19-24.
Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.

Row lengths of A223702.
First differences of A285283.
Cf. A002313, A177979, A185389.

A249132 Smallest noncomposite k such that prime(n) is the largest prime factor of k^2+1, or 0 if no such k exists.

Original entry on oeis.org

1, 0, 2, 0, 0, 5, 13, 0, 0, 17, 0, 31, 73, 0, 0, 23, 0, 11, 0, 0, 173, 0, 0, 233, 463, 293, 0, 0, 251, 919, 0, 0, 37, 0, 193, 0, 443, 0, 0, 599, 0, 19, 0, 467, 211, 0, 0, 0, 0, 107, 89, 0, 659, 0, 241, 0, 2503, 0, 337, 53, 0, 3671, 0, 0

Offset: 1

Juri-Stepan Gerasimov, Oct 22 2014

nonn

a(A080148(m)) = 0. - Joerg Arndt, Oct 22 2014

			a(1)=1 is in this sequence because 1 is in A008578 and the largest prime factor of 1^2+1 = 2 is prime(1).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A080148, A185389, A223702, A223705.

A249132:= proc(n) local p,i,k,a,b;
   p:= ithprime(n);
   if p mod 4 = 3 then return 0 fi;
   a:= numtheory:-msqrt(-1,p);
   if a < p/2 then b:= p-a
   else b:= a; a:= p-a
   fi;
   for i from 0 do
    for k in [a+i*p,b+i*p] do
      if isprime(k) and p = max(numtheory:-factorset(k^2+1)) then
        return(k)
      fi
    od
   od
end proc:
1,seq(A249132(n),n=2..100); # Robert Israel, Nov 10 2014

a249132[n_Integer] := Module[{t = Table[0, {n}], k, s, p}, Do[If[Mod[Prime[k], 4] == 3, t[[k]] = -1], {k, n}]; k = 0; While[Times @@ t == 0, k++; s = FactorInteger[k^2 + 1][[-1, 1]]; p = PrimePi[s]; If[p <= n && t[[p]] == 0 && ! CompositeQ[k], t[[p]] = k]]; t /. -1 -> 0]; a249132[120] (* Michael De Vlieger, Nov 11 2014, adapted from A223702 *)

A379347 a(n) is the sum of all integers of the form k^2 + 1 whose greatest prime factor is A002313(n), the n-th prime not congruent to 3 mod 4.

Original entry on oeis.org

1, 12, 327, 391, 703, 20510, 5667, 661016, 507004, 644098, 24977604, 38394505, 2621510449, 465558141, 624692559, 63435958, 507041846, 8133206945, 70119049516045, 45102364892, 49035127231, 154823547391

Offset: 1

Andrew Howroyd, Dec 22 2024

See A223702 for additional information.

			a(2) = 12 = 2 + 3 + 7. The corresponding values for k^2 + 1 are 5, 10 and 50 each of whose greatest prime factor is 5 = A002313(2).

Row sums of A223702.

Cf. A002313, A185389, A379344.

cp's OEIS Frontend

A223701 Irregular triangle of numbers k such that prime(n) is the largest prime factor of k^2 - 1.

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A285283 Number of integers x such that the greatest prime factor of x^2 + 1 is at most A002313(n), the n-th prime not congruent to 3 mod 4.

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A379350 Triangle read by rows in which row n lists numbers k such that the greatest prime factor of k^2 + 2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

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A379346 Number of integers of the form k^2 + 1 whose greatest prime factor is A002313(n), the n-th prime not congruent to 3 mod 4.

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A249132 Smallest noncomposite k such that prime(n) is the largest prime factor of k^2+1, or 0 if no such k exists.

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A379347 a(n) is the sum of all integers of the form k^2 + 1 whose greatest prime factor is A002313(n), the n-th prime not congruent to 3 mod 4.

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